Investigating limit cycles in nonlinear production and inventory control models

Even in a deterministic setting nonlinearities can yield unexpected dynamic behaviours in an inventory control system, such as rogue oscillations or limit cycles. Utilising a well-known benchmark nonlinear production and inventory control model, we investigate the occurrence of limit cycles.

Design: Nonlinear control theory in combination with simulation is used to analyse the effect of discontinuous nonlinearities present in a production and inventory control model. The describing function method is used to predict limit cycle occurrence and their characteristics, such as frequency, amplitude and stability.

Findings: Findings suggest that, even for an autonomous system, limit cycles occur. This periodic behaviour is observed in the inventory profile when the feedback gain of the WIP is half smaller than the lead-time. Moreover, we demonstrate the potential of the describing function method to accurately predict limit cycle properties.

Value: This paper fills the gap in the literature on nonlinear supply chains by using control engineering methods to explore the dynamic behaviour of inventory profiles. Most studies of supply chain dynamics have focused on linear mathematical models or rely on simulation, which greatly limit the relevancy and/or rigour of published results

Research limitations: This research is limited to the dynamics of a single-echelon supply chain system in a deterministic environment.

Practical implications: The method suggested in this research for analysing nonlinearities in a real-world setting can be used by supply chain designers to gain more insights into nonlinear systems and provide a mechanism to create a set of systematic experiments for simulation rather than relying on a time-consuming ‘trial and error’ approach.